Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-13T06:09:31.384Z Has data issue: false hasContentIssue false
  • English
  • Français

On some generic very cuspidal representations

Part of: Lie groups Linear algebraic groups and related topics

Published online by Cambridge University Press:  18 March 2010

Stephen DeBacker  and
Mark Reeder
Stephen DeBacker
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA (email: smdbackr@umich.edu)
Mark Reeder
Affiliation:
Department of Mathematics, Boston College, Chestnut Hill, MA 02467, USA (email: reederma@bc.edu)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let G be a reductive p-adic group. Given a compact-mod-center maximal torus S⊂G and sufficiently regular character χ of S, one can define, following Adler, Yu and others, a supercuspidal representation π(S,χ) of G. For S unramified, we determine when π(S,χ) is generic, and which generic characters it contains.

Keywords

generic representationvery cuspidalWhittaker modelMoy–Prasad depth

MSC classification

Secondary: 22E50: Representations of Lie and linear algebraic groups over local fields 20G25: Linear algebraic groups over local fields and their integers 20G10: Cohomology theory
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 4 , July 2010 , pp. 1029 - 1055
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Adler, J. D., Refined anisotropic K-types and supercuspidal representations, Pacific J. Math. 185 (1998), 1–32.CrossRefGoogle Scholar
[2]Adler, J. and DeBacker, S., Murnaghan–Kirillov theory for supercuspidal representations of general linear groups, J. Reine Angew. Math. 575 (2004), 1–35.CrossRefGoogle Scholar
[3]Carayol, H., Représentations cuspidales du groupe linéaire, Ann. Sci. Éc. Norm. Sup. 17 (1984), 191–225.CrossRefGoogle Scholar
[4]Carter, R., Finite groups of Lie type (Wiley, New York, 1985).Google Scholar
[5]DeBacker, S., Parametrizing nilpotent orbits via Bruhat–Tits theory, Ann. of Math. (2) 156 (2002), 295–332.CrossRefGoogle Scholar
[6]DeBacker, S., Parametrizing conjugacy classes of maximal unramified tori, Michigan. Math. J. 54 (2006), 157–178.CrossRefGoogle Scholar
[7]DeBacker, S. and Reeder, M., Depth-zero supercuspidal L-packets and their stability, Ann. of Math. (2) 169 (2009), 795–901.CrossRefGoogle Scholar
[8]Gérardin, P., Construction des séries discrètes p-adiques, Lecture Notes in Mathematics, vol. 462 (Springer, Berlin, 1975).CrossRefGoogle Scholar
[9]Harish-Chandra, , Admissible invariant distributions on reductive p-adic groups, University Lecture Series, vol. 16 (American Mathematical Society, Providence, RI, 1999), (Preface and notes by Stephen DeBacker and Paul J. Sally, Jr).CrossRefGoogle Scholar
[10]Howe, R., Tamely ramified supercuspidal representations of GL n, Pacific J. Math. 73 (1977), 437–460.CrossRefGoogle Scholar
[11]Kim, J.-L. and Murnaghan, F., Character expansions and unrefined minimal K-types, Amer. J. Math. 125 (2003), 1199–1234.CrossRefGoogle Scholar
[12]Kostant, B., Lie group representations on polynomial rings, Amer. J. Math. 73 (1977), 437–460.Google Scholar
[13]Kottwitz, R., Course notes (1995).Google Scholar
[14]Kottwitz, R., Transfer factors for Lie algebras, Represent Theory 3 (1999), 127–138 (electronic).CrossRefGoogle Scholar
[15]Langlands, R., Representations of abelian algebraic groups, Pacific J. Math. 61 (1997), 231–250.CrossRefGoogle Scholar
[16]Mœglin, C. and Waldspurger, J.-L., Modèles de Whittaker dégénérées pour des groupes p-adiques, Math. Z. 196 (1987), 427–452.CrossRefGoogle Scholar
[17]Moy, A. and Prasad, G., Unrefined minimal K-types for p-adic groups, Invent. Math. 116 (1994), 393–408.CrossRefGoogle Scholar
[18]Prasad, G., Volumes of S-arithmetic quotients of semi-simple groups, Publ. Math. Inst. Hautes Études Sci. 69 (1989), 91–114.CrossRefGoogle Scholar
[19]Ranga Rao, R., Orbital integrals in reductive groups, Ann. of Math. (2) 96 (1972), 505–510.Google Scholar
[20]Reeder, M., Supercuspidal L-packets of positive depth and twisted Coxeter elements, J. Reine Angew. Math. 620 (2008), 1–33.CrossRefGoogle Scholar
[21]Rodier, F., Whittaker models for admissible representations of reductive p-adic split groups, in Harmonic analysis on homogeneous spaces, Proceedings of Symposia in Pure Mathematics, vol. 26 (American Mathematical Society, Providence, RI, 1973), 425–430.CrossRefGoogle Scholar
[22]Serre, J. P., Local fields (Springer, Berlin, 1979).CrossRefGoogle Scholar
[23]Shelstad, D., Regular elements of semi-simple algebraic groups, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 281–312.Google Scholar
[24]Shelstad, D., A formula for regular unipotent germs, Orbites unipotentes et representations, II, Astérisque 171–172 (1989), 275–277.Google Scholar
[25]Tits, J., Reductive groups over p-adic fields, in Automorphic forms, representations, andL-functions, Proceedings of Symposia in Pure Mathematics, vol. 33, part 1, eds Borel, A. and Casselman, W. (American Mathematical Society, Providence, RI, 1979), 29–69.CrossRefGoogle Scholar
[26]Yu, J.-K., Construction of tame supercuspidal representations, J. Amer. Math. Soc. 14 (2001), 579–622.CrossRefGoogle Scholar